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Properties

Label 2-459-153.106-c1-0-9
Degree 22
Conductor 459459
Sign 0.451−0.892i0.451 - 0.892i
Analytic cond. 3.665133.66513
Root an. cond. 1.914451.91445
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.92 + 1.10i)2-s + (1.45 + 2.52i)4-s + (0.680 + 0.182i)5-s + (2.07 − 0.556i)7-s + 2.04i·8-s + (1.10 + 1.10i)10-s + (−1.34 + 0.360i)11-s + (0.789 + 1.36i)13-s + (4.60 + 1.23i)14-s + (0.657 − 1.13i)16-s + (−0.767 + 4.05i)17-s − 0.618i·19-s + (0.532 + 1.98i)20-s + (−2.98 − 0.798i)22-s + (−0.903 + 3.37i)23-s + ⋯
L(s)  = 1  + (1.35 + 0.784i)2-s + (0.729 + 1.26i)4-s + (0.304 + 0.0815i)5-s + (0.784 − 0.210i)7-s + 0.721i·8-s + (0.349 + 0.349i)10-s + (−0.405 + 0.108i)11-s + (0.218 + 0.379i)13-s + (1.23 + 0.329i)14-s + (0.164 − 0.284i)16-s + (−0.186 + 0.982i)17-s − 0.141i·19-s + (0.119 + 0.444i)20-s + (−0.635 − 0.170i)22-s + (−0.188 + 0.703i)23-s + ⋯

Functional equation

Λ(s)=(459s/2 ΓC(s) L(s)=((0.451−0.892i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.451 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(459s/2 ΓC(s+1/2) L(s)=((0.451−0.892i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 459459    =    33⋅173^{3} \cdot 17
Sign: 0.451−0.892i0.451 - 0.892i
Analytic conductor: 3.665133.66513
Root analytic conductor: 1.914451.91445
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ459(208,⋅)\chi_{459} (208, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 459, ( :1/2), 0.451−0.892i)(2,\ 459,\ (\ :1/2),\ 0.451 - 0.892i)

Particular Values

L(1)L(1) ≈\approx 2.61403+1.60785i2.61403 + 1.60785i
L(12)L(\frac12) ≈\approx 2.61403+1.60785i2.61403 + 1.60785i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
17 1+(0.767−4.05i)T 1 + (0.767 - 4.05i)T
good2 1+(−1.92−1.10i)T+(1+1.73i)T2 1 + (-1.92 - 1.10i)T + (1 + 1.73i)T^{2}
5 1+(−0.680−0.182i)T+(4.33+2.5i)T2 1 + (-0.680 - 0.182i)T + (4.33 + 2.5i)T^{2}
7 1+(−2.07+0.556i)T+(6.06−3.5i)T2 1 + (-2.07 + 0.556i)T + (6.06 - 3.5i)T^{2}
11 1+(1.34−0.360i)T+(9.52−5.5i)T2 1 + (1.34 - 0.360i)T + (9.52 - 5.5i)T^{2}
13 1+(−0.789−1.36i)T+(−6.5+11.2i)T2 1 + (-0.789 - 1.36i)T + (-6.5 + 11.2i)T^{2}
19 1+0.618iT−19T2 1 + 0.618iT - 19T^{2}
23 1+(0.903−3.37i)T+(−19.9−11.5i)T2 1 + (0.903 - 3.37i)T + (-19.9 - 11.5i)T^{2}
29 1+(2.71+10.1i)T+(−25.1+14.5i)T2 1 + (2.71 + 10.1i)T + (-25.1 + 14.5i)T^{2}
31 1+(6.01+1.61i)T+(26.8+15.5i)T2 1 + (6.01 + 1.61i)T + (26.8 + 15.5i)T^{2}
37 1+(2.82−2.82i)T−37iT2 1 + (2.82 - 2.82i)T - 37iT^{2}
41 1+(−0.573+2.14i)T+(−35.5−20.5i)T2 1 + (-0.573 + 2.14i)T + (-35.5 - 20.5i)T^{2}
43 1+(−4.30−2.48i)T+(21.5+37.2i)T2 1 + (-4.30 - 2.48i)T + (21.5 + 37.2i)T^{2}
47 1+(−4.53+7.85i)T+(−23.5−40.7i)T2 1 + (-4.53 + 7.85i)T + (-23.5 - 40.7i)T^{2}
53 1−12.8iT−53T2 1 - 12.8iT - 53T^{2}
59 1+(−5.64+3.25i)T+(29.5−51.0i)T2 1 + (-5.64 + 3.25i)T + (29.5 - 51.0i)T^{2}
61 1+(2.85−0.765i)T+(52.8−30.5i)T2 1 + (2.85 - 0.765i)T + (52.8 - 30.5i)T^{2}
67 1+(5.68+9.83i)T+(−33.5+58.0i)T2 1 + (5.68 + 9.83i)T + (-33.5 + 58.0i)T^{2}
71 1+(−5.47+5.47i)T−71iT2 1 + (-5.47 + 5.47i)T - 71iT^{2}
73 1+(−4.12+4.12i)T−73iT2 1 + (-4.12 + 4.12i)T - 73iT^{2}
79 1+(7.43−1.99i)T+(68.4−39.5i)T2 1 + (7.43 - 1.99i)T + (68.4 - 39.5i)T^{2}
83 1+(2.68+1.55i)T+(41.5+71.8i)T2 1 + (2.68 + 1.55i)T + (41.5 + 71.8i)T^{2}
89 1−8.02T+89T2 1 - 8.02T + 89T^{2}
97 1+(4.51+16.8i)T+(−84.0+48.5i)T2 1 + (4.51 + 16.8i)T + (-84.0 + 48.5i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−11.47401743215686876018130758272, −10.49977837013861379326900212522, −9.385723478968790951158652065925, −8.045004932605389128719014156310, −7.42965489996910369379145656128, −6.22912310332799484723772855290, −5.61383821549695642246494884355, −4.50631421608278635994374135295, −3.73995211176008239367703585853, −2.08699429884232938178016636614, 1.71582785641283146746764518306, 2.85744824333333144559321225305, 4.01240951659053631229838383996, 5.19811452462364304998447018547, 5.56355666360427030368731860429, 7.00117605332359937190421560101, 8.202219726853843043967538393722, 9.244764318747041127456703220128, 10.49949464985746467935989886814, 11.08366611472603341204853046084

Graph of the ZZ-function along the critical line